Feedback Classification of linear systems and convolutional codes. Applications in cybernetics, coding theory and cryptography

Résumé

Several natural phenomena are mathematically modeled through linear systems of differential equations. We study the feedback classification of linear systems over commutative rings with identity. In particular, we characterize von Neumann regular rings in terms of linear systems. Moreover, we give an explicit formula to give the number of classes of feedback isomorphisms of locally Brunovsky linear systems with state space X over different commutative rings by partitions in a monoid. Furthermore, we show that we can explicit interconnections between control theory and coding over certain commutative rings by generalizing the duality that exists over finite fields. We also study equivalence relations between certain families of convolutional codes and its I/S/O representations by their Kronecker Indices and their conjugate partitions. This final result is given by the correspondences between classes of feedback isomorphisms of locally Brunovsky linear system and classes of feedback isomorphisms of dynamical behaviours of I/S/O representations of certain families of convolutional codes. Finally, we give an overview of some areas of research with which this work is related. Some of them are a field of future research and the other areas can be considered as a field of implementation and applications